This might come a bit surprising for many, but the Men In Black organization was a secretive division of SHIELD. One obvious clue is that Marvel owns rights to the Men In Black comic book series. Marvel acquired rights to the series when it bought Malibu Comics in 1994.
Men In Black was part of a universe of its own. It had no connection to the Marvel Universe. However, when Marvel acquired the rights to the comic book series, Men In Black automatically joined the Marvel Universe. Marvel has a tendency to buy rights to comic book characters and fighting them into the Marvel shared universe. Sometimes the storylines do not make sense, but they try to do it somehow. In the case of Men In Black comics, there is no issue where another Marvel character appeared in a Men In Black storyline. Now that Men In Black film series is highly popular, we might expect it to join the Marvel Cinematic Universe. This is only if Marvel/Disney could negotiate with Sony Pictures who currently owns the rights to movie franchises. Marvel already did it with the Spider-Man series; maybe they could do it again.
The organization formed in Men in Black in the mid-1950s roughly the same time when SHIELD was formally established. Initially, The Men in Black was a small, underfunded government organization with the main objective of making contact with extraterrestrial life known as Baltians. They eventually made contact on March 2, 1961, just outside New York City. Afterward, it was decided to make Earth an apolitical zone for aliens without a plane. Basically, they wanted to make the Earth safe heaven for alien refugees. It broke ties with the government. It became the agency to police and to conceal the aliens who came to Earth. It does not answer to any government and does not oblige with outside demands. However, it may assist governments in matters regarding aliens. MIB is funded by patents they hold on confiscated alien technology. MIB also protects the Earth from extraterrestrial threats.
If MIB to fit into the Marvel Cinematic Universe, it would be a division of SHIELD that splintered away and became an independent agency. SHIELD is a government-owned and controlled organization. SHIELD might receive assistant from MIB from time to time. For example, in Captain Marvel, it is shown that a Kree scientist has been working for them all this time. However, SHIELD agents such as Nick Fury did not know that aliens even existed before the events of Captain Marvel. In Captain Marvel, we see that SHIELD technology is the same as what an average person would use. Only high tech devices we saw were the ones developed by Dr. Wendy Lawson, who happens to be an alien. After the events of Captain Marvel, we see that SHIELD tech has considerably advanced compared to the rest of the world. We might assume, SHIELD got a little help from MIB.
I hope Marvel/Disney will do a crossover with MIB in their cinematic universe in the future. Hopefully we will get some form of reference to this in Men in Black: International. As a joke, Spider-Man: Far From Home could be titled Spider-Man: International.